Prove this block matrices are similar

Prove that the block matrices
$
\left(
\begin{array}{cc}
AB & 0\\
B & 0\\
\end{array}
\right)
$
and
$
\left(
\begin{array}{cc}
0 & 0\\
B & BA\\
\end{array}
\right)
$
are similar.

Where $\mathbf{K}$ is any field, $A\in \mathbf{K}^{m\times n}$, $B\in \mathbf{K}^{n\times m}$
and both matrices in $\mathbf{K}^{(m+n)\times (m+n)}$.

I searched the Internet well enough and found no similar problem.

Thanks in advance!

Solutions Collecting From Web of "Prove this block matrices are similar"

From Fuzhen Zhang, “Matrix Theory – Basic Results and Techniques” (Springer, 1999), page 52:

\begin{align*}
\begin{bmatrix} {\rm I}_m & A \\ 0 & {\rm I}_n \end{bmatrix} \begin{bmatrix} 0 & 0 \\ B & 0 \end{bmatrix} = \begin{bmatrix} AB & 0 \\ B & 0 \end{bmatrix}, \\
\begin{bmatrix} 0 & 0 \\ B & 0 \end{bmatrix} \begin{bmatrix} {\rm I}_m & A \\ 0 & {\rm I}_n \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ B & BA \end{bmatrix}, \\
\end{align*}
follows that
$$\begin{bmatrix} {\rm I}_m & A \\ 0 & {\rm I}_n \end{bmatrix}^{-1} \begin{bmatrix} AB & 0 \\ B & 0 \end{bmatrix} \begin{bmatrix} {\rm I}_m & A \\ 0 & {\rm I}_n \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ B & BA \end{bmatrix}.$$